Minimal configuration unicyclic graphs

Abstract

Abstract A graph G is singular with nullity η( G ), if zero is an eigenvalue of its adjacency matrix with multiplicity η( G ). If η(G) = 1, then the core of G is the subgraph induced by the vertices associated with the non-zero entries of the kernel eigenvector. The set of vertices which are not in the core is called the periphery of G. A graph G with nullity one is called a minimal configuration if no two vertices in the periphery are adjacent and deletion of any vertex in the periphery increases the nullity. In this article, we describe the structure of a singular unicyclic graph and single out the class of unicyclic graphs which are minimal configurations. Keywords: Adjacency matrixSingular graphsCoreMinimal configurationUnicyclic graph 2000 Mathematics Subject Classifications: : 05C5005C6005B20